Modelling multiple sample selection in intergenerational occupational mobility Cheti Nicoletti ISER, University of Essex Marco Francesconi Department of.

Modelling multiple sample selection in intergenerational occupational mobility

Cheti NicolettiISER, University of Essex

Marco FrancesconiDepartment of Economics, University of Essex

Main aims of the paper

1. Estimation of intergenerational occupational mobility in Britain.

2. Correcting for potential sample selection problems in short panels using different estimation methods.

iii iAxy '

Sample selection problems

• Labour market selection: Intergenerational occupational mobility can be estimated only for people who are employed.

• Coresidence selection: Children must be living together with their parents in at least one wave of the panel.

1991 2003

BHPS 1991-1993

Child age

3

8

13

18

23

28

33

Child age

15

20

25

30

35

44

45

1997

Child age

9

14

19

24

29

34

39

Cohort

1988

1983

1978

1973

1968

1963

1958

Taking account of coresidence selection

Francesconi and Nicoletti (2006) find that the intergenerational mobility in occupational prestige is underestimated when using the subsample of sons born between 1966 and 1985.

They try different estimation methods to correct for sample selection and find that only the inverse propensity score is able to attenuate the selection problem

This sample selection evaluation is possible because all BHPS respondents are asked to report occupational characteristics of their parents when they were 14

Taking account of selection into employment for children

• Blanden (2005) and Ermisch et al (2005) consider two-step estimation procedures and find lower and unchanged βs

• Couch and Lillard (1998) and Nicoletti and Francesconi (2006) consider imputation methods and find lower βs

• Minicozzi (2003) use partial identification approach to produce bound estimates instead than point estimates for the intergenerational mobility and find higher βs when including unemployed and part-time workers.

Contributions of the paper

Propose new estimation methods to take account of sample selection problem in the intergenerational mobility models which are very parsimonious

Taking account of both coresidence and employment selection bias

Selection models

If εi and ui are not independent then we have selection due to unobservables

If εi depends on Zi then we have selection due to observables

If εi depends on Zi and ui then we have selection due to both observables and unobservables

iii xy

iuZdii

'*)0( *

iidId where

Selection due to unobservables

y=α+xβ+ε d*=Z γ+u d=l(d*>0)

Let E(ε|x)=0, ε ind Z, (ε, u) be N with means zeros, variances σ2 and 1 and covariance ρ

Then E((y-α-xβ) |x,d=1) ≠ 0 and OLSE is biased E(y|x,d=1)=α+xβ+E(ε|x,d=1)=α+xβ+ ρλ v=ε- ρλ is such that E(v|d=1,X)=0 We can consider an additional correction term

(Heckman 1979, Vella 1998)

Selection due to observables

y=α+xβ+ε d*=Z γ+u d=l(d*>0)

Let E(ε|x)=0, ε ╨ u but ε not ind Z Then E(ε|x,d=1)≠0 and OLSE is biased

because of selection on observables Since ε ╨ d|x,Z we can adopt (1) propensity score

methods, (2) regression adjustment methods or (3) combining methods. (see Rosembaum and Rubin, 1983; Robins and Rotnitzky, 1995; Hirano et al., 2003)

Propensity score weighting method

Let Pr(d=1|x,Z)=Pr(d=1|Z)=p(Z)

Then E(ε d|x) ≠ 0 but E(ε d p(Z)-1|x)=0E(ε d p(Z)-1|x)= EZE(ε d p(Z)-1|x,Z)

= EZ[E(ε |x,Z,d=1) Pr(d=1|x,Z)p(Z)-1]

Since ε ╨ d|x,Z

= EZ[E(ε |x,Z) Pr(d=1|x,Z)p(Z)-1]

=EZ[E(ε |x,Z)]=E(ε |x)=0

This holds even if some of the variables in Z are erroneously omitted from the main equation.

Regression adjustmenty=α+xβ+ε d*=Z γ+u d=l(d*>0)

• To take account that ε is not ind of Z

y=αN+xβN+Zδ+ω• If the linearity assumption is satisfied then

E(ω|X,Z,d=1)=E(ω|X,Z)=E(ω|X)=0 and• βN is consistently estimated• β=Cov(x,y)/Var(x)=βN+Cov(x,Z)Var(Z)-1δ

Combining regression adjustment and propensity score method

Estimation of the extended model y=α+xβ+Zδ+ωby using inverse propensity score weighting

E[(y-α-xβ-Zδ) d p(Z)-1|x]= EZE[(y-α-xβ- Zδ) d p(Z)-1|x,Z]= EZ[E(y-α-xβ- Zδ |x,Z,d=1) Pr(d=1|x,Z)p(Z)-1]

Notice that this expression is 0if either E(y-α-xβ- Zδ |x,Z,d=1)=E(ω|X,Z,d=1)=0 or Pr(d=1|x,Z)=p(Z) holds and not necessarilyboth.

Selection due to both observables and unobservables

y=α+xβ+ε d*=Z γ+u d=l(d*>0)

where ε depends on both Z and u

(ε, u) is N with means zeros, variances σ2 and 1 and

covariance ρ

v=(ε- ρλ) ind d |x,Z

We can use: (1) Heckman correction and propensity

score weighting or (2) Heckman correction and regression adjustment.

Heckman correction & propensity score weighting

E[(y-α-xβ- ρλ) d p(Z)-1|x]

= EZE[(y-α-xβ- ρλ) d p(Z)-1|x,Z]

= EZ[E(y-α-xβ- ρλ|x,Z,d=1) Pr(d=1|x,Z)p(Z)-1]

Since (y-α-xβ- ρλ) ╨ d|x,Z

= EZ[E(y-α-xβ- ρλ |x,Z) Pr(d=1|x,Z)p(Z)-1]

=EZ[E(y-α-xβ- ρλ |x,Z)]= E(y-α-xβ- ρλ |x)= 0

Heckman correction & regression adjustment

Estimation of the extended model with

additional variables Z and correction term λ

y=α+xβ+Zδ+ ρλ +ω

d*=Z γ+u

• ρλ controls for the dependence of ε1 on u

• Zδ controls for the dependence of ε2 on Z

How can the BHPS help us?All BHPS respondents are asked to report occupational

characteristics of their parents when they were 14THEREFORE

• We know the occupational prestige even for daughters and fathers living apart during the panel.

• We can estimate the intergenerational mobility without any coresidence selection.

• We can consider the subsample of daughters coresident with the fathers at least once during the panel and assess the relevance of the coresidence selection.

• We can then compare different methods to correct for the coresidence selection.

BHPS Samples

• FULL SAMPLE: 2691 women (daughters) born between 1966 and 1985 with at least one valid interview over the first 13 waves of the BHPS (aged between 16-37, average 24)

• RESTRICTED SAMPLE: 745 individuals from the full sample who can be matched with their father (aged between 16-37, average age 21).

• We consider an average occupational prestige over all waves available for daughters. We consider instead the occupation prestige reported retrospectively by daughters for fathers (average age 46).

Estimation Methods Used

• Inverse propensity score weighting (Weights)

• Regression adjustment

• Regression adjustment & weights

• Heckman correction method (Heckman)

• Heckman & weights

• Regression adjustment & Heckman

Coresidence selection model

y=α+xβ+Aμ+ε d*=Z γ+u d=l(d*>0)where y is the daughter’s occupational prestige (log Hope-

Goldthorpe score) x is her father’s occupational prestige A age and age2

d=1 for daughters living together with their father in at least one wave and 0 otherwise

Z=dummies for education, age, regions, ethnicity, religiosity and two house price indexes

The intergenerational equation is too parsimonious

y=α+xβ+Aμ+ε d*=Z γ+u d=l(d*>0)

Education dummies are important to explaining both the daughters occupational prestige and their probability to be coresident

The assumption that ε ╨ d is not acceptable.

Regression adjustment when x is missing

y=αN+xβN+Zδ+ω

• If the linearity assumption is satisfied

• βN is consistently estimated

• β=Cov(x,y)/Var(x)=βN+Cov(x,Z)Var(Z)-1δ

• If x is missing it is not possible to estimate Cov(x,Z) consistently

Correcting for coresidence selection only

β SE

Full sample 0.250 0.028

Restricted sample 0.147 0.044

Weights 0.208 0.084

Heckman 0.145 0.043

Heckman and weights 0.206 0.083

Regression adjustment 0.135 0.043

Regression adjustment & Heckman 0.132 0.043

Regression adjustment & weights 0.206 0.063

Employment selection model

y=α+xβ+Aμ+ε d*=Z γ+u d=l(d*>0)where y is the daughter’s occupational prestige (log Hope-

Goldthorpe score) x is her father’s occupational prestige A age and age2

d=1 for daughters are employed at least in at least one wave and 0 otherwise

Z=occupation prestige father, dummies for education, age, regions, ethnicity, religiosity, a house price index, marital status and number of children aged between 0-2, 3-4, 5-11, 12-15, 16-18.

Correcting for employment selection only

β SE

Full sample 0.250 0.028

Weights 0.265 0.031

Heckman 0.209 0.041

Heckman and weights 0.227 0.032

Regression adjustment 0.249 0.028

Regression adjustment & Heckman 0.255 0.029

Regression adjustment & weights 0.253 0.030

Correcting for employment and sample selection simultaneously

β SE

Full sample 0.250 0.028

Restricted sample 0.147 0.044

Weights Bivariate selection 0.208 0.084

Regression adj & weights Bivariate selection 0.145 0.043

Weights 0.206 0.083

Regression adjustment & weights 0.135 0.043

Heckman 0.132 0.043

Heckman & Regression adjustment 0.132 0.043

Heckman & weights 0.206 0.063

Matching selection in quantile regressions

Quantile Full Restricted Weights

10 0.386 0.125 0.384

0.069 0.100 0.133

25 0.257 0.219 0.281

0.052 0.077 0.131

50 0.248 0.164 0.215

0.063 0.071 0.109

75 0.240 0.109 0.138

0.045 0.054 0.096

90 0.079 0.059 0.141

0.033 0.070 0.067

Employment selection in quantile regressions

Quantile Full Weights

10 0.386 0.311

0.069 0.070

25 0.257 0.266

0.052 0.046

50 0.248 0.292

0.063 0.049

75 0.240 0.279

0.045 0.041

90 0.079 0.195

0.033 0.040

Double selection in quantile regressions

Quantile Full Restricted Weights

10 0.386 0.125 0.389

0.069 0.100 0.132

25 0.257 0.219 0.389

0.052 0.077 0.136

50 0.248 0.164 0.244

0.063 0.071 0.146

75 0.240 0.109 0.109

0.045 0.054 0.131

90 0.079 0.059 0.008

0.033 0.070 0.112

Conclusions

• The intergenerational equation is too parsimonious and there are probably omitted variables such as education dummies.

• In this situation correcting for selection on observables is much more important than correcting for selection on unobservables.

• The coresidence selection seems to cause an underestimation of β.

• The selection into employment does not seem to cause a large bias in β.